I completely understand this question has been addressed one too many times. But I still simply cannot wrap my head around the concept of dy/dx.
Simply put, when can we treat dy/dx as a ratio and when can we not in physics? I have been time and again told dy/dx is NOT a ratio but a limit of a difference quotient, if that be the case, I keep seeing classical mechanics authors indiscriminately treat it like a fraction!
For instance, I have just recently seen this:
Given position vector r(t), where r(t) = sin(t) i + cos(t) j (i and j are unit vectors along x and y axis.
Then I suddenly see that dr(t) = (cot(t) i - sin(t) j) dt. How? Why? What? I have made such computations when I was in high-school, but I never thought about such things. But now that I am, I find it most disconcerting.
How is this working? This is in context of the line integral with regards to work done. The author here is discussing the line integral of f.ds
Also, how can we ever take the dot product of an infinitesimal with the force. The ds, after all, and in the Riemann integral sense, is just a delimiter, and otherwise has no other computational value. I have been told that that it is USEFUL to view ds as an infinitesimal breadth, but I don't want to do things because they are a means to an end, I want to fully understand the means, no matter how trivial.
I really, really, want to understand the truth of dy/dx, so that I may be able to use the tool to 'build' and 'craft' some mathematics. But as of now, I am just using the tool without any insight.
I came in to discuss and unrelated problem hours ago in this site and N hours after i'm still here. But saw your question come in and I actually find this enchanting (sorry it got down voted). But I wanted to share some of my thinking in this area, and maybe we can give a different perspective to this issue that's so mentioned in physics education!
Anyway, I think what physics textbooks should actually cover is the idea of "implicit differentiation", see https://www.mathsisfun.com/calculus/implicit-differentiation.html . It is correct to say that $dy/dx$ is not a fraction, it is a guide telling us what limits we are looking at. And it just so happens that when we have comparable "guides" then this syntax does look a tad like a fraction.
A simplified version of your example, if $r = r(t) = \sin t$, then $$ \frac{dr}{dt} = \frac{d}{dt} \sin t = \cos t \frac{dt}{dt} = \cos{t} $$ However, if we looked at it as an exercise of implicit differentiation the above could be seen as $$ dr = \cos{t} dt $$
Which comes into play because then you can do line integrals properly as the above inside an integral would be seen as $$ \int dr = r = \int \cos (t) dt = \sin{t} + C $$
Note how the first integral is now being done along $r$, which allows us to solve $\int dx = x + c$, instead of being puzzled over how $r$ depends on some other variables.
tl;dr knowing that implicit differentiation is mathematically valid and about doing integrals with the simplest set of variables I think is what physics textbooks should talk about instead of giving us a riddle about something not being some other thing without telling us what it actually is (implicit differentiation, at least most of the time).